A field has no ideal, since this is it not a field it might.
(Note: I worked on this problem before when you posted it and it got a little messy by looking at all the possible ideal. What I was doing was finding all the cyclic subgorups of this ring because they need to be cyclic for the additive group is cylcic)

Is this correct?

No

And could someone just explain quickly what a homomorphic image is?

I think you mean when you have a ring homomorphism the image of is a homomorphic image.